The study of generalized quasi–Einstein manifolds is introduced in [1]}, which is a natural generalization of the Einstein metrics. More precisely, an \(n\) -dimensional complete Riemannian manifold \(M\)  is a generalized quasi–Einstein manifold if there exist three smooth functions \(f\) , \(\mu \) , and \(\lambda \)  on \(M\)  satisfying
\(Ric_{f}^{\, \mu } = \lambda g.\) 
